Theorem frgrancvvdeqlem3 30793

Description: Lemma 3 for frgrancvvdeq 30803. In a friendship graph, for each neighbor of a vertex there is exacly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van der Vekens, 22-Dec-2017.)

Hypotheses

Ref	Expression
frgrancvvdeq.nx	|- D = ( <. V , E >. Neighbors X )
frgrancvvdeq.ny	|- N = ( <. V , E >. Neighbors Y )
frgrancvvdeq.x	|- ( ph -> X e. V )
frgrancvvdeq.y	|- ( ph -> Y e. V )
frgrancvvdeq.ne	|- ( ph -> X =/= Y )
frgrancvvdeq.xy	|- ( ph -> Y e/ D )
frgrancvvdeq.f	|- ( ph -> V FriendGrph E )
frgrancvvdeq.a	|- A = ( x e. D |-> ( iota_ y e. N { x , y } e. ran E ) )

Assertion

Ref	Expression
frgrancvvdeqlem3	|- ( ( ph /\ x e. D ) -> E! y e. N { x , y } e. ran E )

Distinct variable groups:   y, D   x, y, V   x, E, y   y, Y   ph, y

Allowed substitution hints:   ph( x)   A( x, y)   D( x)   N( x, y)   X( x, y)   Y( x)

Proof of Theorem frgrancvvdeqlem3

Step	Hyp	Ref	Expression
1		frgrancvvdeq.f	. . . 4 |- ( ph -> V FriendGrph E )
2	1	adantr 465	. . 3 |- ( ( ph /\ x e. D ) -> V FriendGrph E )
3		frgrancvvdeq.nx	. . . . . . 7 |- D = ( <. V , E >. Neighbors X )
4	3	eleq2i 2532	. . . . . 6 |- ( x e. D <-> x e. ( <. V , E >. Neighbors X ) )
5		frisusgra 30752	. . . . . . 7 |- ( V FriendGrph E -> V USGrph E )
6		nbgraisvtx 23514	. . . . . . 7 |- ( V USGrph E -> ( x e. ( <. V , E >. Neighbors X ) -> x e. V ) )
7	1, 5, 6	3syl 20	. . . . . 6 |- ( ph -> ( x e. ( <. V , E >. Neighbors X ) -> x e. V ) )
8	4, 7	syl5bi 217	. . . . 5 |- ( ph -> ( x e. D -> x e. V ) )
9	8	imp 429	. . . 4 |- ( ( ph /\ x e. D ) -> x e. V )
10		frgrancvvdeq.y	. . . . 5 |- ( ph -> Y e. V )
11	10	adantr 465	. . . 4 |- ( ( ph /\ x e. D ) -> Y e. V )
12		frgrancvvdeq.ny	. . . . . 6 |- N = ( <. V , E >. Neighbors Y )
13		frgrancvvdeq.x	. . . . . 6 |- ( ph -> X e. V )
14		frgrancvvdeq.ne	. . . . . 6 |- ( ph -> X =/= Y )
15		frgrancvvdeq.xy	. . . . . 6 |- ( ph -> Y e/ D )
16		frgrancvvdeq.a	. . . . . 6 |- A = ( x e. D |-> ( iota_ y e. N { x , y } e. ran E ) )
17	3, 12, 13, 10, 14, 15, 1, 16	frgrancvvdeqlem1 30791	. . . . 5 |- ( ( ph /\ x e. D ) -> Y e. ( V \ { x } ) )
18		eldif 3449	. . . . . 6 |- ( Y e. ( V \ { x } ) <-> ( Y e. V /\ -. Y e. { x } ) )
19		ssnid 4017	. . . . . . . . 9 |- x e. { x }
20		eleq1 2526	. . . . . . . . . 10 |- ( Y = x -> ( Y e. { x } <-> x e. { x } ) )
21	20	eqcoms 2466	. . . . . . . . 9 |- ( x = Y -> ( Y e. { x } <-> x e. { x } ) )
22	19, 21	mpbiri 233	. . . . . . . 8 |- ( x = Y -> Y e. { x } )
23	22	necon3bi 2681	. . . . . . 7 |- ( -. Y e. { x } -> x =/= Y )
24	23	adantl 466	. . . . . 6 |- ( ( Y e. V /\ -. Y e. { x } ) -> x =/= Y )
25	18, 24	sylbi 195	. . . . 5 |- ( Y e. ( V \ { x } ) -> x =/= Y )
26	17, 25	syl 16	. . . 4 |- ( ( ph /\ x e. D ) -> x =/= Y )
27	9, 11, 26	3jca 1168	. . 3 |- ( ( ph /\ x e. D ) -> ( x e. V /\ Y e. V /\ x =/= Y ) )
28		frgraunss 30755	. . 3 |- ( V FriendGrph E -> ( ( x e. V /\ Y e. V /\ x =/= Y ) -> E! y e. V { { x , y } , { y , Y } } C_ ran E ) )
29	2, 27, 28	sylc 60	. 2 |- ( ( ph /\ x e. D ) -> E! y e. V { { x , y } , { y , Y } } C_ ran E )
30		prex 4645	. . . . . . . . . . . 12 |- { x , y } e. _V
31		prex 4645	. . . . . . . . . . . 12 |- { y , Y } e. _V
32	30, 31	prss 4138	. . . . . . . . . . 11 |- ( ( { x , y } e. ran E /\ { y , Y } e. ran E ) <-> { { x , y } , { y , Y } } C_ ran E )
33		simpr 461	. . . . . . . . . . 11 |- ( ( { x , y } e. ran E /\ { y , Y } e. ran E ) -> { y , Y } e. ran E )
34	32, 33	sylbir 213	. . . . . . . . . 10 |- ( { { x , y } , { y , Y } } C_ ran E -> { y , Y } e. ran E )
35	34	ad2antll 728	. . . . . . . . 9 |- ( ( ( ph /\ x e. D ) /\ ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) ) -> { y , Y } e. ran E )
36	12	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> N = ( <. V , E >. Neighbors Y ) )
37	36	eleq2d 2524	. . . . . . . . . . . 12 |- ( ph -> ( y e. N <-> y e. ( <. V , E >. Neighbors Y ) ) )
38		nbgraeledg 23513	. . . . . . . . . . . . 13 |- ( V USGrph E -> ( y e. ( <. V , E >. Neighbors Y ) <-> { y , Y } e. ran E ) )
39	1, 5, 38	3syl 20	. . . . . . . . . . . 12 |- ( ph -> ( y e. ( <. V , E >. Neighbors Y ) <-> { y , Y } e. ran E ) )
40	37, 39	bitrd 253	. . . . . . . . . . 11 |- ( ph -> ( y e. N <-> { y , Y } e. ran E ) )
41	40	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ x e. D ) -> ( y e. N <-> { y , Y } e. ran E ) )
42	41	adantr 465	. . . . . . . . 9 |- ( ( ( ph /\ x e. D ) /\ ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) ) -> ( y e. N <-> { y , Y } e. ran E ) )
43	35, 42	mpbird 232	. . . . . . . 8 |- ( ( ( ph /\ x e. D ) /\ ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) ) -> y e. N )
44		simpl 457	. . . . . . . . . 10 |- ( ( { x , y } e. ran E /\ { y , Y } e. ran E ) -> { x , y } e. ran E )
45	32, 44	sylbir 213	. . . . . . . . 9 |- ( { { x , y } , { y , Y } } C_ ran E -> { x , y } e. ran E )
46	45	ad2antll 728	. . . . . . . 8 |- ( ( ( ph /\ x e. D ) /\ ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) ) -> { x , y } e. ran E )
47	43, 46	jca 532	. . . . . . 7 |- ( ( ( ph /\ x e. D ) /\ ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) ) -> ( y e. N /\ { x , y } e. ran E ) )
48	47	ex 434	. . . . . 6 |- ( ( ph /\ x e. D ) -> ( ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) -> ( y e. N /\ { x , y } e. ran E ) ) )
49	12	eleq2i 2532	. . . . . . . . . . . . 13 |- ( y e. N <-> y e. ( <. V , E >. Neighbors Y ) )
50	49, 39	syl5bb 257	. . . . . . . . . . . 12 |- ( ph -> ( y e. N <-> { y , Y } e. ran E ) )
51	50	biimpd 207	. . . . . . . . . . 11 |- ( ph -> ( y e. N -> { y , Y } e. ran E ) )
52	51	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ x e. D ) -> ( y e. N -> { y , Y } e. ran E ) )
53	52	impcom 430	. . . . . . . . 9 |- ( ( y e. N /\ ( ph /\ x e. D ) ) -> { y , Y } e. ran E )
54		nbgraisvtx 23514	. . . . . . . . . . . . . . . 16 |- ( V USGrph E -> ( y e. ( <. V , E >. Neighbors Y ) -> y e. V ) )
55	1, 5, 54	3syl 20	. . . . . . . . . . . . . . 15 |- ( ph -> ( y e. ( <. V , E >. Neighbors Y ) -> y e. V ) )
56	49, 55	syl5bi 217	. . . . . . . . . . . . . 14 |- ( ph -> ( y e. N -> y e. V ) )
57	56	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. D ) -> ( y e. N -> y e. V ) )
58	57	impcom 430	. . . . . . . . . . . 12 |- ( ( y e. N /\ ( ph /\ x e. D ) ) -> y e. V )
59	58	ad2antlr 726	. . . . . . . . . . 11 |- ( ( ( { y , Y } e. ran E /\ ( y e. N /\ ( ph /\ x e. D ) ) ) /\ { x , y } e. ran E ) -> y e. V )
60		simpl 457	. . . . . . . . . . . . 13 |- ( ( { y , Y } e. ran E /\ ( y e. N /\ ( ph /\ x e. D ) ) ) -> { y , Y } e. ran E )
61		id 22	. . . . . . . . . . . . 13 |- ( { x , y } e. ran E -> { x , y } e. ran E )
62	60, 61	anim12ci 567	. . . . . . . . . . . 12 |- ( ( ( { y , Y } e. ran E /\ ( y e. N /\ ( ph /\ x e. D ) ) ) /\ { x , y } e. ran E ) -> ( { x , y } e. ran E /\ { y , Y } e. ran E ) )
63	62, 32	sylib 196	. . . . . . . . . . 11 |- ( ( ( { y , Y } e. ran E /\ ( y e. N /\ ( ph /\ x e. D ) ) ) /\ { x , y } e. ran E ) -> { { x , y } , { y , Y } } C_ ran E )
64	59, 63	jca 532	. . . . . . . . . 10 |- ( ( ( { y , Y } e. ran E /\ ( y e. N /\ ( ph /\ x e. D ) ) ) /\ { x , y } e. ran E ) -> ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) )
65	64	ex 434	. . . . . . . . 9 |- ( ( { y , Y } e. ran E /\ ( y e. N /\ ( ph /\ x e. D ) ) ) -> ( { x , y } e. ran E -> ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) ) )
66	53, 65	mpancom 669	. . . . . . . 8 |- ( ( y e. N /\ ( ph /\ x e. D ) ) -> ( { x , y } e. ran E -> ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) ) )
67	66	impancom 440	. . . . . . 7 |- ( ( y e. N /\ { x , y } e. ran E ) -> ( ( ph /\ x e. D ) -> ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) ) )
68	67	com12 31	. . . . . 6 |- ( ( ph /\ x e. D ) -> ( ( y e. N /\ { x , y } e. ran E ) -> ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) ) )
69	48, 68	impbid 191	. . . . 5 |- ( ( ph /\ x e. D ) -> ( ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) <-> ( y e. N /\ { x , y } e. ran E ) ) )
70	69	eubidv 2285	. . . 4 |- ( ( ph /\ x e. D ) -> ( E! y ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) <-> E! y ( y e. N /\ { x , y } e. ran E ) ) )
71	70	biimpd 207	. . 3 |- ( ( ph /\ x e. D ) -> ( E! y ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) -> E! y ( y e. N /\ { x , y } e. ran E ) ) )
72		df-reu 2806	. . 3 |- ( E! y e. V { { x , y } , { y , Y } } C_ ran E <-> E! y ( y e. V /\ { { x , y } , { y , Y } } C_ ran E ) )
73		df-reu 2806	. . 3 |- ( E! y e. N { x , y } e. ran E <-> E! y ( y e. N /\ { x , y } e. ran E ) )
74	71, 72, 73	3imtr4g 270	. 2 |- ( ( ph /\ x e. D ) -> ( E! y e. V { { x , y } , { y , Y } } C_ ran E -> E! y e. N { x , y } e. ran E ) )
75	29, 74	mpd 15	1 |- ( ( ph /\ x e. D ) -> E! y e. N { x , y } e. ran E )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   E!weu 2262   =/= wne 2648   e/ wnel 2649   E!wreu 2801   \ cdif 3436   C_ wss 3439   {csn 3988   {cpr 3990   <.cop 3994   class class class wbr 4403   |-> cmpt 4461   ran crn 4952   iota_crio 6163  (class class class)co 6203   USGrph cusg 23436   Neighbors cnbgra 23501   FriendGrph cfrgra 30748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-fz 11558  df-hash 12224  df-usgra 23438  df-nbgra 23504  df-frgra 30749

This theorem is referenced by:  frgrancvvdeqlem4  30794  frgrancvvdeqlem5  30795